What is No Solution for Linear Equations?

S7-SA2-0041

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

When we try to solve a system of linear equations and find that there is no value for the variables (like 'x' and 'y') that can satisfy all the equations at the same time, we say it has 'no solution'. This usually happens when the equations represent parallel lines that never intersect.

Simple Example

Quick Example

Imagine you have two friends, Priya and Rohan. Priya says, 'I am exactly 5 years older than Rohan.' Rohan says, 'I am exactly 7 years younger than Priya.' If you try to find their ages, you'll see a problem. Priya being 5 years older means P = R + 5. Rohan being 7 years younger means R = P - 7, which is P = R + 7. These two statements contradict each other, so there's no possible age for them that satisfies both!

Worked Example

Step-by-Step

Let's try to solve this system of linear equations:
Equation 1: x + y = 5
Equation 2: x + y = 7
---Step 1: We want to find values for 'x' and 'y' that make both equations true.
---Step 2: Look at Equation 1: x + y equals 5.
---Step 3: Now look at Equation 2: x + y equals 7.
---Step 4: Can the exact same 'x' and 'y' add up to 5 AND also add up to 7 at the same time? No, that's impossible!
---Step 5: If we try to subtract Equation 1 from Equation 2, we get:
(x + y) - (x + y) = 7 - 5
0 = 2
---Step 6: Since 0 can never be equal to 2, this means there is no solution for 'x' and 'y' that satisfies both equations simultaneously.
Answer: No Solution.

Why It Matters

Understanding 'no solution' is crucial in fields like AI/ML for training models, in Physics for predicting system behaviors, and in Engineering for designing stable structures. Engineers use this concept to identify impossible conditions in designs, ensuring safety and efficiency in everything from bridge construction to electric vehicle batteries.

Common Mistakes

MISTAKE: Assuming there must always be a solution for any system of equations. | CORRECTION: Always check if the equations are consistent. If they lead to a contradiction (like 0 = 5), then there is no solution.

MISTAKE: Confusing 'no solution' with 'many solutions' (infinite solutions). | CORRECTION: 'No solution' means no common point exists. 'Many solutions' means the equations are actually the same line, so all points on that line are solutions.

MISTAKE: Making calculation errors and concluding 'no solution' incorrectly. | CORRECTION: Double-check all steps, especially when substituting or subtracting equations, to ensure the contradiction is real and not a mistake.

Practice Questions

Try It Yourself

QUESTION: Does the system x + 2y = 10 and x + 2y = 12 have a solution? | ANSWER: No, it has no solution.

QUESTION: Find the value of 'k' for which the following system of equations has no solution: 3x + 4y = 5 and 6x + 8y = k. | ANSWER: For no solution, the ratio of coefficients should be equal (a1/a2 = b1/b2) but not equal to the ratio of constants (c1/c2). Here, 3/6 = 4/8 = 1/2. So, 5/k should not be 1/2. This means k should not be 10. Any value of k except 10 will lead to no solution. (Specifically, if k=10, it's infinite solutions. If k is anything else, it's no solution.)

QUESTION: A vendor sells samosas and jalebis. On Monday, he says, 'I sold 20 items in total, and the number of samosas was 5 more than the number of jalebis.' On Tuesday, he says, 'I sold 20 items in total, and the number of samosas was 7 more than the number of jalebis.' Can both statements be true for Monday and Tuesday respectively? Why or why not? | ANSWER: For Monday, let S be samosas and J be jalebis. S + J = 20 and S = J + 5. For Tuesday, S + J = 20 and S = J + 7. Both statements for Monday are consistent (S=12.5, J=7.5 - though not whole numbers, it's mathematically possible). Both statements for Tuesday are also consistent (S=13.5, J=6.5). However, the question asks if BOTH 'number of samosas was 5 more than jalebis' AND 'number of samosas was 7 more than jalebis' can be true for the SAME day (which is implied by 'Can both statements be true'). No, they cannot. A number cannot be 5 more than another AND 7 more than it at the same time. This would lead to a contradiction like 5 = 7, meaning no solution.

MCQ

Quick Quiz

Which of the following conditions indicates that a system of two linear equations in two variables has no solution?

The lines intersect at exactly one point.

The lines are identical (overlap each other).

The lines are parallel and distinct (do not intersect).

The lines pass through the origin.

The Correct Answer Is:

When lines are parallel and distinct, they never meet, meaning there is no common point (no solution) that satisfies both equations. Intersecting lines have one solution, and identical lines have infinitely many solutions.

Real World Connection

In the Real World

Imagine you're a traffic engineer trying to optimize traffic flow in a busy Indian city. You set up equations to model the number of cars on two different roads. If your equations for the car flow on two flyovers lead to a 'no solution' scenario, it means your current plan for diverting traffic creates an impossible situation – perhaps two traffic streams are expected to occupy the same space at the same time without collision, which is impossible. You'd need to redesign the traffic plan to find a 'solution' that works.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation whose graph is a straight line, like y = mx + c. | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables. | PARALLEL LINES: Lines that are always the same distance apart and never intersect. | INCONSISTENT SYSTEM: A system of equations that has no solution.

What's Next

What to Learn Next

Now that you understand 'no solution', you should explore systems with 'unique solutions' and 'infinitely many solutions'. This will give you a complete picture of all possible outcomes when solving linear equations, which is super important for higher-level math and science!